Questions tagged [young-tableaux]
For standard Young tableaux, semistandard Young tableaux, and other related two-dimensional arrays of numbers like plane partitions. Including their combinatorial theory and their application in representation theory and algebraic geometry.
185 questions
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Formula for sum involving products of (symplectic) Schur functions
This question is a continuation of a question asked yesterday which had a very nice answer.
Consider the summation
$$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
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Number of semistandard tableaux of all possible shapes fitting within some rectangle
Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum
$$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$
where $S_\lambda$ denotes the Schur ...
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Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
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Combine two types of permutations in a Young diagram?
Given a Young diagram $Y$, for each row $R$ choose a permutation $\sigma_R$ of $\{1,\dots, |R|\}$, where $|R|$ is the size of row $R$. Let $\sigma_R(i)$ be the “row coordinate” of the $i$th cell in ...
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Map between Weyl modules in terms of Young tableaux
The irreducible algebraic representations of $\text{GL}_n$ over the complex numbers are given by highest weight representations of dominant weights $\lambda=(k_1,k_2,\ldots,k_n): k_1 \ge k_2 \ge \...
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Generating function for A225114
Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns).
Let $b(n)$ be an integer sequence with generating function $B(x)$ such that
$$
B(x) = \...
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Why $f^\lambda$ in the hook-length formula?
This is my first question on this site so I apologize if it’s not adequate for it.
I just learned the hook-length formula for the number $f^\lambda$ of Standard Young Tableaux of shape $\lambda$:
$$f^\...
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Proving an identity for flagged Schur without use of determinants?
In proposition 3 of Determinantal transition kernels for some interacting particles on the line, Dieker and Warren prove the following identity: consider vector $a:=(a_1,\dotsc,a_N)$ and kernels
$$\...
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Describing the hook part of the symmetric group algebra
Let $\mathbf{k}$ be a field of characteristic $0$. Let $n\in\mathbb{N}$, and
consider the symmetric group $S_{n}$ consisting of all permutations of
$\left[ n\right] :=\left\{ 1,2,\ldots,n\right\} $...
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Garnir elements basic question
Prop. 2.6.3 in Sagan's book The symmetric group discusses Garnir elements, and says:
"If $|A\cup B|$ is greater than the number of elements in column $j$ of $t$, then $g_{A,B} \mathbf{e}_t=0$.&...
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A problem about the existence of increasing coloring groups
Got stuck on this one for months.
Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...
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Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram
I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$:
\begin{equation}
d_\lambda = \sum_{a \in \mathrm{...
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Schur functors = Weyl functors in characteristic zero?
I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
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Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$
Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
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Errata for Fulton's "Young tableaux"
Fulton's Young tableaux is one of the best texts on the subject, one which I
often recommend and cite for reference. Unlike Fulton/Lang and
Fulton/Harris,
it is neither an early-dawn draft nor a ...
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geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety
For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For ...
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Skewed plane partition with only row fillings reversed
The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
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Orthogonality of irreducible and non-isomorphic representations [closed]
Let V and W be any two subspaces of $(\mathbb{C}^d)^{\otimes n}$ such that there exists two irreducible and non-isomorphic representations $\rho_V: G \to GL(V)$ and $\rho_W: G \to GL(W)$. Does this ...
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Decomposition of tensors into symmetry classes according to Schur functors
I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree.
As it is well-known and extremely easy to ...
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Counting certain kinds of Semistandard Young Tableaux
We have a project in which it is natural to count the number of Young Tableau in which part of the weight has been specified. Does anybody know if this idea already appears in the literature?
More ...
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Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as
$$
V(\pi_1) \...
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Generalized Gaussian binomial and symmetric chain decomposition
Background
Let $\mu = (\mu_1, \ldots, \mu_k)$ be a partition, meaning that $\mu_1 \geq \ldots \mu_k \geq 1$. The Young diagram associated to $\mu$ is given by the set $(r,c) \in \mathbb{N} \times \...
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hook length formula for plane partitions
The hook length formula give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ...
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Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule
$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
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Mistakes in Logan and Shepp's famous paper on Young Tableaux?
In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...
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References for applications of Young diagrams/tableaux to Quantum Mechanics
I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book:
Wybourne, B.G.; "...
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Young tableaux — irreps correspondence for simple complex Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally
introduced to study the irreducible representations of finite
symmetric groups $S_n$ ...
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A formula for the generating function of Hoggatt binomials or of some Young tableaux
Let ${\left\langle\matrix {n \cr k}\right\rangle}_r$ denote the $r-$Hoggatt binomials defined by
$${{\left\langle\matrix {n \cr k}\right\rangle}_r=\frac{\langle n \rangle_r!}{\langle k \rangle_r! \...
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Robinson-Schensted-Knuth (RSK) under restriction
I am curious about the following result concerning the Robinson-Schensted insertion procedure. I can formulate a proof via the Schützenberger evacuation operator, but I have struggled to find such an ...
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Counting integer partitions below some Young diagram
Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the ...
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Bijection between forests and skew SYT + Cyclic sieving
Consider the two-row skew shape $\lambda_n = (2n+1,n)/(1)$.
The number of standard Young tableaux of this shape is
$\binom{3n}{n}-\binom{3n}{n-2}$ (since one can easily biject this to the set of non-...
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On a proof involving Young symmetrizers acting on tensor spaces
I hope this is not too elementary for this site, but I already asked something similar on MSE which has not received any attention whatsoever. I am extremely unfamiliar with the algebraic/...
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$\mathfrak{sl}_2$-action on Young diagrams
Let $\mathcal{Y}$ be a vector $\mathbb{Q}$-space of all Young diagrams. Denote by $\delta_\lambda$ the Young diagram of the partition $\lambda$ and $c(\square)$ be
the content of the square $\...
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The Grassmann twist-map, an associated semi-group action, and RSK
Let me begin by setting some notation: Let $\mathrm{Mat}_{k,n}(\Bbb{R})$ denote the vector space of all $k \times n$
real-valued matrices. Given $g \in \mathrm{Mat}_{k,n}(\Bbb{R})$ and two (ordered) ...
- 2,137
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Hives for other root systems? [duplicate]
Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...
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Relationship between $\mathbb{S}^{\nu}V \otimes \mathbb{S}^{\lambda}(V^{*})$ and $\mathbb{S}^{\nu / \lambda}V$
For partition $\mu$ let $\mathbb{S}^{\mu}V = V^{\otimes \mu} \cdot c_{\mu}$, where $c_{\mu}$ is the Young symmetrizer. I'm trying to prove that $\mathbb{S}^{\nu / \lambda}V$ is the polynomial part of $...
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Addition theorem for Schur function in multivariable
Working with the following problem Expansion in Schur function of negative binomial exponent
I need to find the expansion of
$$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$
in terms of schur ...
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Yamanouchi ribbon tableaux?
Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions.
The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ ...
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Status of conjecture of Conrey and Gonek, combinatorial meaning
I was looking at the OEIS on the number of square Young Tableaux.
In it Michael Somos referenced a paper of Conrey and Gonek, High Moment's of the Riemann Zeta-Function. Is there an combinatorial ...
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Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
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Number of paths to a specific vertex in the Young's lattice
Consider the Young's lattice. What is the number of paths starting from the origin (0) to a specific Young diagram?
For instance, the Young diagram corresponding to the integer partition 1+1+1 has 1 ...
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Number of branches between two layers of the Young's lattice
In the Young's lattice, the number of branches that connect the $N$'th layer to the $N+1$'th layer has the sequence:
$$
1,2, 4, 7, 12, 19, 30, 45, 67, 97, 139, \cdots
$$
Looking this up on OEIS, leads ...
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Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule
The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
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Map between irreducible representations in basis given by Young tableaux
Let $V$ be a $n$-dimensional complex vector space.
Assume we have a $\mathbb C$-linear map $\varphi:\Gamma^{(a_1,\dots,a_n)}V\rightarrow \Gamma^{(b_1,\dots,b_n)}V$ between two irreducible ...
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LGV scheme: Any situations where the weights shift differently for each path?
In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder
In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...
- 5,474
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Two majs for standard Young tableaux?
Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
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LGV scheme for lattice paths that move in non-unit spatial positive steps
In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here).
What do we mean by "time"? In the language of ...
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A question related to Young symmetrizers
Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...
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The distribution of amajor over standard Young tableaux?
Given a standard Young tableau $T$, $i$ is called an ascent of $T$ if $i+1$ is in a higher row than $i$, and amajor$(T)$ is defined to be the sum of ascents of $T$. For the distribution of the amajor ...
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RSK correspondence for sum of two matrices
The celebrated RSK correspondence (see Wikipedia page) assigns to each integer $X$ matrix a pair of Young tableaux $P$ and $Q$. Now suppose that we have three integer matrices $X_1$, $X_2$, and $X_3$, ...
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